Smooth atlases

Let $M$ be a topological $n$-manifold. A smooth atlas on $M$ is a family $\mathcal{A}=(U_i,{\phi }_i{)}_{i\in I}$ of charts on $M$ (Manifold charts, coordinate domains, and coordinate functions) such that:

1. the coordinate domains cover $M$, that is ${\bigcup }_{i\in I}{U}_i=M$; and
2. any two members of $\mathcal{A}$ are smoothly compatible (Smoothly compatible charts and the smoothness of Euclidean transition maps).

Two atlases $\mathcal{A}$ and $\mathcal{B}$ on $M$ are compatible when every chart of $\mathcal{A}$ is smoothly compatible with every chart of $\mathcal{B}$. The family of all charts of both atlases is written $\mathcal{A}\cup \mathcal{B}$.

• The covering condition is part of being an atlas. A family of pairwise compatible charts that does not cover $M$ is not an atlas, and the maximal atlas of Each smooth atlas is contained in a unique maximal smooth atlas still covers $M$ for exactly this reason.

• Compatibility of atlases is cross-pairwise, not pairwise within the union. Compatibility within each atlas is already given; the extra content of "compatible atlases" is the smoothness of every transition across the two families, which is the hypothesis The union of two compatible smooth atlases is a smooth atlas turns into an atlas again.